Just a Quick Question

If I were to ask you to multiply 99 x 3.

Would you do the calculation in your head? 99 x 3 = 97

Most everybody I asked would do the (head math) 300 – 3 = 97.

I have no knowledge of algebra and asked a friend one night over many
adult beverages if he could help me figure out the thought process used
above in an algebraic equation. Since we couldn’t find a symbol to represent
the “next higher multiple of 10 we settled for Ҁ.

As you can see below we are woefully incapable of such a task.

This has probably been thought out hundreds of years ago but I’ve not been
able to find a solution to my question.

Please be kind to a 64 year old guy who flunked algebra twice in high school.

ps. When I looked at the equation below the next day I could only laugh.
It's all Greek to me!

Re: Just a Quick Question

Originally Posted by ChuckB

If I were to ask you to multiply 99 x 3.

Would you do the calculation in your head? 99 x 3 = 97

Most everybody I asked would do the (head math) 300 – 3 = 97.

I have no knowledge of algebra and asked a friend one night over many
adult beverages if he could help me figure out the thought process used
above in an algebraic equation. Since we couldn’t find a symbol to represent
the “next higher multiple of 10 we settled for Ҁ.

As you can see below we are woefully incapable of such a task.

This has probably been thought out hundreds of years ago but I’ve not been
able to find a solution to my question.

Please be kind to a 64 year old guy who flunked algebra twice in high school.

ps. When I looked at the equation below the next day I could only laugh.
It's all Greek to me!

Re: Just a Quick Question

Skill at doing mental arithmetic does not require a great deal of intelligence: it requires knowing a few methods that simplify arithmetic and a well trained memory. It used to be a skill that was useful to those who worked with numbers a lot, but today with calculators sitting on your phone, it has far less practical use.

An example of a method for simplifying problems.

18 X 17

First step: Both are close to 20. Remember 20, 18, and 17.

Second step: 20 - 18 = 2. Remember 20, 2, and 17.

Third step: 20 - 17 = 3. Remember 20, 2, and 3.

Fourth step: 20 X 20 = 400. Remember 400, 2, and 3.

Fifth step: 2 + 3 = 5. Remember 400, 2, 3, and 5.

Sixth step: 5 X 20 = 100. Remember 400, 100, 2, and 3.

Seventh step: 400 - 100 = 300. Remember 300, 2, and 3.

Eighth step: 2 X 3 = 6. Remember 300 and 6.

Nineth step: 300 + 6 = 306. Answer.

Each step is very simple arithmetic, and you never need remember no more than 4 numbers at a time.

Why does it work?

$(a - b)\ X\ (a - c) =$

$\{a\ X\ (a - c)\} - \{b\ X\ (a - c)\} =$

$(a\ X\ a) - (a\ X\ c) - (a\ X\ b) + (b\ X\ c) =$

$(a\ X\ a) - \{a\ X\ (b + c)\} + (b\ X\ c).$

Here are two longish articles about the topic of rapid mental arithmetic. The first discusses records. The second methods.

A = 99
B = 3
Ҁ = 100 (it's 100 as it would be the next higher multiple of 10 from 99)

Now, if you plug in all these values, then 'C' comes out to the correct value of 297. Of course the funny thing is the [A Ҁ – (A Ҁ –A)] portion works out to be 99, thus you are back to 99 x 3 which of course is 297.

I believe this equation was a bit of an exercise to put into a formula what we were thinking intuitively in our minds. Intuitively we know that 99 x 3 is the same as (100 x 3) - 3, but how would you describe that exercise as a formula? Explaining in words would involve saying:
"Go up to the nearest 10 of the largest number, multiply it by the small number, then subtract the amount that you added to the large number by the small number."

We also had to invent a special symbol that we called 'Ҁ', as there is nothing in math that represents the "going up to the nearest 10 of a number".

Hope this helps explain what is going on here. We were basically messing around with taking an explanation of something normally done with words, and putting it into a formula. A bit of a brain exercise.

Another way to view it would be to know that:
99 x 3 = ((99 + 1) x 3) - (1 x 3)